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Purely Periodic beta-Expansions in the Pisot Non-unit Case

Valerie Berthe 1 Anne Siegel 2
1 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
2 SYMBIOSE - Biological systems and models, bioinformatics and sequences
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : It is well known that real numbers with a purely periodic decimal expansion are rationals having, when reduced, a denominator coprime with $10$. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit. We characterize real numbers having a purely periodic expansion in such a base. This characterization is given in terms of an explicit set, called a generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and $p$-adic spaces.
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Submitted on : Wednesday, October 24, 2007 - 5:43:39 PM
Last modification on : Friday, February 4, 2022 - 3:16:33 AM
Long-term archiving on: : Monday, April 12, 2010 - 12:28:50 AM


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  • HAL Id : inria-00181997, version 1


Valerie Berthe, Anne Siegel. Purely Periodic beta-Expansions in the Pisot Non-unit Case. Journal of Number Theory, Elsevier, 2007, 127 (2), pp.153-172. ⟨inria-00181997⟩



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